The position vectors of four points \(A, B, C, D\) relative to an origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\) respectively. Obtain the equation of the line \(l\) joining the midpoints of \(AB\) and \(CD\). Show that \(l\) intersects the line joining the midpoints of \(AC\) and \(BD\) and find the position vector \(\mathbf{p}\) of the point \(P\) of intersection.

Deduce that the line joining the midpoints of \(AD\) and \(BC\) also passes through \(P\).